This equation is valid for any real or complex values of $f_0,~f_1,~a,~\&~b$. Integer sequences accrue only when all of them are integers, however.

Now, when the radical in $\alpha,\beta$ is real, so are $\alpha,\beta$ and the limit of consecutive quotients is given by

$$\lim_{n\to\infty}\frac{f_{n+1}}{f_n}=\alpha$$

However, when that radical is imaginary, then $\alpha,\beta$ are complex conjugates and both $F_n$ and $\frac{f_{n+1}}{f_n}$ are oscillatory. Nevertheless, the sequences are real. (Many such sequences can be found in the OEIS.) To see how this arises, consider

Similarly, for the simplified case when $f_0=0$, the limit of consecutive quotients is given by
$$
\begin{align}
\lim_{n\to\infty}\frac{f_{n+1}}{f_n}
&=\lim_{n\to\infty}\frac{F_{n+1}}{F_n}\\
&=\frac{\mathfrak{Im}\{\alpha^{n+1}\}}{\mathfrak{Im}\{\alpha^n\}}
=|\alpha|\frac{\sin (n+1)\theta}{\sin n\theta}\\
\end{align}
$$

The figure below shows a typical limit ratio versus $n$. The behavior is seen to be oscillatory, but not periodic. It almost looks chaotic, what with quasi-repeating patterns and several obvious frequencies.

I'm seek help to characterize this behavior. I have tried treating it as a 'signal' but did not get very far with either the FFT or HHT (Hilbert-Huang transform), though I readily admit to having not used either in several years.

$\begingroup$What is the question? You already have a very clean closed formula: what else do you want to know?$\endgroup$
– Qiaochu YuanSep 4 '17 at 21:17

$\begingroup$@QiaochuYuan I want to be able characterize the figure in the sense of determining the characteristic frequencies, or spacing between the peaks, and how they relate to the input parameters. As I said, treat it like a signal. What can you tell me about it. Having the equation allows me to plot it, but doesn't tell me anything else about it.$\endgroup$
– Cye WaldmanSep 4 '17 at 22:58

$\begingroup$The closed formula tells you all sorts of things about it. For starters, note that if $\theta$ is a rational multiple of $\pi$ then the closed formula implies that the sequence is periodic. If $\theta$ is close to some rational multiple of $\pi$ then it is close to periodic, as you've observed, so the question of what the periodicities look like is straightforwardly tied to the question of how to approximate $\frac{\theta}{\pi}$ by rationals.$\endgroup$
– Qiaochu YuanSep 5 '17 at 4:04